The memristor device postulated in 1971 by Leon Chua [1] as the fourth basic circuit element has received much attention in the research community since the publication of Strukov's 2008 paper titled “The missing memristor found” [2]. The memristor name is a contraction for memory resistor [1] because that is exactly its function: to remember its history [3]. The memristor is a two terminal passive device whose resistance state depends on its previous state and present electrical biasing conditions. And combined with transistors in a hybrid chip, memristors could radically improve the performance of digital circuits without the necessity to shrink transistors [3]. Given their two terminal structural simplicity and electronic passivity, the applications for memristor technology range from non-volatile memory, instant on computers, reconfigurable electronics and neuromorphic computing [4],[3]. According to Chua [4], the memristor behaves like a linear resistor with memory but exhibits many interesting nonlinear characteristics, and several electronic models have been presented to describe the electrical behavior of memristor devices [1],[4],[2],[5],[6]. However, given that memristor devices are not commercially available, good physical model to hardware methods have not yet been reported m the published literature. Therefore, methods for accurate memristor modeling is an impending need.
Prior Art Memristor Models
Referring to FIG. 1 depicts the typical DC electrical Lissajous I-V curve characteristic response of a memristor device measured under test with a sinusoidal input of 0.5 V volts amplitude and 100 Hz frequency. From the figure we can clearly observe that the memristor device toggles between two states: high and low conductivity. As the memristor device transitions from a low conductivity state to a high conductivity state it exhibits highly nonlinear diode-like characteristics at approximately −0.35 and 0.2 V threshold voltages, respectively. that switched the device from a low conductivity state to a high conductivity state and vice-versa. The threshold voltage analogy is used here to describe the voltage biasing region where nonlinear behavior occurs.
Referring to FIG. 2, the most basic mathematical definition of a memristor is that of a current-controlled device for circuit analysis in the generalized class of nonlinear dynamical systems called memristive systems described by the equations
                    v        =                              R            ⁡                          (                              w                ,                i                            )                                ⁢          i                                    (        1        )                                                      ⅆ            w                                ⅆ            t                          =                  f          ⁡                      (                          w              ,              i                        )                                              (        2        )            where w can be a set of state variables and R and f can in general be explicit functions of time [2],[4]. For simplicity and ease of simulation, the memristor's resistance or “memristance” can be represented as a current-controlled. time-invariant, one-port device given by
                              M          ⁡                      (            w            )                          =                                            w              D                        ⁢                          R              on                                +                                    (                              1                -                                  w                  D                                            )                        ⁢                          R              off                                                          (        3        )            where represents the doped region of the memristor, D the total length of the memristor device, Ron the lowest rest stance state and R,rr the highest resistance state as graphically described [2].
In order to describe the velocity at which w increases. meaning the rate at which the memristor device is becoming less resistive, Equation 2 can be described as follows
                                          ⅆ                          w              ⁡                              (                t                )                                                          ⅆ            t                          =                              u            v                    ⁢                                    R              on                        D                    ⁢                      i            ⁡                          (              t              )                                                          (        4        )            where uv is the average ion mobility for the simplest case of ohmic electronic conduction and linear ionic drift m a uniform field [2].
Utilizing Ohm's law that states that the voltage across a resistor is directly proportional to the resistance times the current through the conductor; we can obtain from equations (3) and (4) the following relationship
                              w          ⁡                      (            t            )                          =                              u            v                    ⁢                                    R              on                        D                    ⁢                      q            ⁡                          (              t              )                                                          (        5        )            and by inserting Equation (5) into Equation (3), we can obtain the memristance of the system, which for an Ron much less than Roff can be simplified to
                              M          ⁡                      (            q            )                          =                              R            off                    ⁡                      (                          1              -                                                u                  v                                ⁢                                                      R                    on                                    D                                ⁢                                  q                  ⁡                                      (                    t                    )                                                                        )                                              (        6        )            [2]. Equation (6) describes the memristance of the memristor system as a function of the charge q(t).
Additional improvements have been proposed to the aforementioned memristor model to include of non-linear boundary conditions [5],[6]. The non-linear boundary condition proposed is of the form
                                          f            p                    ⁡                      (            w            )                          =                  1          -                                    (                                                2                  ⁢                                                                          ⁢                                      w                    D                                                  -                1                            )                                      2              ⁢                                                          ⁢              p                                                          (        7        )            The nonlinear window function in Equation (7) guarantees zero velocity of the doped/undoped barrier interface described in FIG. 1 as w approaches either boundary, w=0 or w=D. Moreover, the differences between the models with linear and nonlinear drift disappear when p increases [5]. The incorporation of the window' function described by Equation (7) requires the redefining of Equation (4) as follows
                                          ⅆ                          w              ⁡                              (                t                )                                                          ⅆ            t                          =                              u            v                    ⁢                                    R              on                        D                    ⁢                      i            ⁡                          (              t              )                                ⁢                                    f              p                        ⁡                          (              w              )                                                          (        8        )            for p=1, it is possible to integrate equation (8) analytically; however, for values of p larger than 1 only numerical solutions are possible [6].
Failure of Prior Art Linear and Non-Linear Memristor Models
Attempts have been made to perform a model fit utilizing both the linear [2] and nonlinear [5],[6] memristor models. Referring to FIG. 3 shows prior art memristor model to hardware correlation fit utilizing the memristor linear model [2]. From the figure, we can observe that the linear model does not accurately capture the high memristor nonlinearities located at approximately the −0.35 and 0.2 V threshold voltages. In particular, and still referring to FIG. 3, there appear to be two regions of memristor operation between −0.35 to −0.05 V and between 0.05 and 0.2 V which the linear memristor model fails to model accurately. FIG. 4 describes the theoretical I-V curves for a memristor device with (realistic) dopant drift modeled by window functions obtained by Joglekar et. al [6]. From the figure, it is clear that the prior art nonlinear model results do not accurately describe the actual DC electrical Lissajous I-V characteristic behavior of physical memristor device hardware.